International J.Math. Combin. Vol.2(2018), 24-32

Equitable Coloring on Triple Star Graph Families

K.Praveena (Department of Computer Science, Dr.G.R. Damodaran College of Science, Coimbatore-641014, Tamilnadu, India)

M.Venkatachalam (Department of Mathematics, Kongunadu Arts and Science College, Coimbatore-641029, Tamilnadu, India)

E-mail: [email protected], [email protected]

Abstract: An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color class differ by at most one. In this paper we investigate the equitable chromatic number for the Central graph, Middle graph, Total graph and Line

graph of Triple star graph K1,n,n,n denoted by C(K1,n,n,n), M(K1,n,n,n), T (K1,n,n,n) and

L(K1,n,n,n) respectively. Key Words: Equitable coloring, Smarandachely equitable k-coloring, triple star graph, central graph, middle graph, total graph and line graph. AMS(2010): 05C15, 05C78.

§1. Introduction

A graph consist of a vertex set V (G) and an edge set E(G). All Graphs in this paper are finite, loopless and without multiple edges. We refer the reader [8] for terminology in graph theory. Graph coloring is an important research problem [7, 10]. A proper k-coloring of a graph is a labelling f : V (G) 1, 2, , k such that the adjacent vertices have different labels. The → { · · · } labels are colors and the vertices with same color form a color class. The chromatic number of a graph G, written as χ(G) is the least k such that G has a proper k-coloring. Equitable colorings naturally arise in some scheduling, partitioning and load balancing problems [11,12]. In 1973, Meyer [4] introduced first the notion of equitable colorability. In 1998, Lih [5] surveyed the progress on the equitable coloring of graphs. We say that a graph G = (V, E) is equitably k-colorable if and only if its vertex set can be partitioned into independent sets V , V , , V V such that V V 1 holds for { 1 2 · · · k} ⊂ || i| − | j || ≤ every pair (i, j). The smallest integer k for which G is equitable k-colorable is known as the equitable chromatic number [1,3] of G and denoted by χ=(G). On the other hand, if V can be partitioned into independent sets V , V , , V V with V V 1 holds for every { 1 2 · · · k}⊂ || i| − | j || ≥ pair (i, j), such a k-coloring is called a Smarandachely equitable k-coloring.

In this paper, we find the equitable chromatic number χ=(G) for central, line, middle and total graphs of triple star graph.

1Received September 14, 2017, Accepted May 15, 2018. Equitable Coloring on Triple Star Graph Families 25

§2. Preliminaries

For a given graph G = (V, E) we do a operation on G, by subdividing each edge exactly once and joining all the non adjacent vertices of G. The graph obtained by this process is called central graph of G [1] and is denoted by C(G). The line graph [6] of a graph G, denoted by L(G) is a graph whose vertices are the edges of G and if u, v E(G) then uv E(L(G)) if u and v share a vertex in G. ∈ ∈ Let G be a graph with vertex set V (G) and edge set E(G). The middle graph [2] of G denoted by M(G) is defined as follows. The vertex set of M(G) is V (G) E(G) in which two ∪ vertices x, y are adjacent in M(G) if the following condition hold:

(1) x, y E(G) and x, y are adjacent in G; ∈ (2) x V (G), y E(G) and they are incident in G. ∈ ∈ Let G be a graph with vertex set V (G) and edge set E(G). The total graph [1,2] of G is denoted by T (G) and is defined as follows. The vertex set of T (G) is V (G) E(G). Two ∪ vertices x, y in the vertex set of T (G) is adjacent in T (G), if one of the following holds:

(1) x, y are in V (G) and x is adjacent to y in G; (2) x, y are in E(G) and x, y are adjacent in G; (3) x is in V (G), y is in E(G) and x, y are adjacent in G.

Triple star K1,n,n,n [9] is a tree obtained from the double star [2] K1,n,n by adding a new pendant edge of the existing n pendant vertices. It has 3n +1 vertices and 3n edges.

§3. Equitable Coloring on Central Graph of Triple Star Graph

Algorithm 1.

Input: The number ‘n’ of k1,n,n,n;

Output: Assigning equitable colouring for the vetices in C(K1,n,n,n).

begin for i = 1to n

{ V = e 1 { i} C(ei)= i; V = a ; 2 { i} C(ai)= i;

} V = v ; 3 { } C(v)= n + 1; 26 K.Praveena and M.Venkatachalam

for i =2 to n

{ V = v ; 4 { i} C(v )= i 1; i − V = w ; 5 { i} C(w )= i 1; i − V = u ; 6 { i} C(u )= i 1; i − } C(v1)= n;

C(w1)= n;

C(u1)= n; for i =1to5

{ V = s ; 7 { i} C(si)= n + 1;

} for i =6 to n

{ V = s ; 8 { i} C(si)= i;

} V = V V V V V V V V ; 1 ∪ 2 ∪ 3 ∪ 4 ∪ 5 ∪ 6 ∪ 7 ∪ 8 end

Theorem 3.1 For any triple star graph K1,n,n,n the equitable chromatic number

χ=[C(K1,n,n,n)] = n +1.

Proof Let v : 1 i n , w : 1 i n and u : 1 i n be the vertices in { i ≤ ≤ } { i ≤ ≤ } { i ≤ ≤ } K . The vertex v is adjacent to the vertices v (1 i n). The vertices v (1 i n) is 1,n,n,n i ≤ ≤ i ≤ ≤ adjacent to the vertices w (1 i n) and the vertices w (1 i n) is adjacent to the vertices i ≤ ≤ i ≤ ≤ u (1 i n). i ≤ ≤ By the definition of central graph on K , let the edges vv , v w and w u (1 i n) 1,n,n,n i i i i i ≤ ≤ of K be subdivided by the vertices e ,a ,s (1 i n) respectively. 1,n,n,n i i i ≤ ≤ Equitable Coloring on Triple Star Graph Families 27

Clearly,

V [C(K )] = v v :1 i n w :1 i n 1,n,n,n { } { i ≤ ≤ } { i ≤ ≤ } u[:1 i n e[:1 i n { i ≤ ≤ } { i ≤ ≤ } [ a :1 i n [ s :1 i n { i ≤ ≤ }∪{ i ≤ ≤ } [ The vertices v and u (1 i n) induces a clique of order n + 1 (say k ) in [C[K ]]. i ≤ ≤ n+1 1,n,n,n Therefore χ [C(K )] n +1 = 1,n,n,n ≥ Now consider the vertex set V [C(K )] and the color class C = c ,c ,c , c . Assign 1,n,n,n { 1 2 3 · · · n+1} an equitable coloring to C(K1,n,n,n) by Algorithm 1. Therefore

χ [C(K )] n +1. = 1,n,n,n ≤ An easy check shows that v v 1. Hence || i| − | j || ≤

χ=[C(K1,n,n,n)] = n +1. ¾

§4. Equitable Coloring on Line graph of Triple Star Graph

Algorithm 2.

Input: The number ‘n’ of K1,n,n,n;

Output: Assigning equitable coloring for the vertices in L(K1,n,n,n).

begin for i =1 to n

{ V = e ; 1 { i} C(ei)= i; V = s ; 2 { i} C(si)= i;

} for i =2 to n

{ V = a ; 3 { i} C(a )= i 1; i − } 28 K.Praveena and M.Venkatachalam

C(a1)= n; V = V V V ; 1 ∪ 2 ∪ 3 end

Theorem 4.1 For any triple star graph K1,n,n,n the equitable chromatic number,

χ=[L(K1,n,n,n)] = n.

Proof Let v : 1 i n , w : 1 i n and u : 1 i n be the vertices in { i ≤ ≤ } { i ≤ ≤ } { i ≤ ≤ } K . The vertex v is adjacent to the vertices v (1 i n) with edges e (1 i n). The 1,n,n,n i ≤ ≤ i ≤ ≤ vertices v (1 i n) is adjacent to the vertices w (1 i n) with edges a (1 i n). The i ≤ ≤ i ≤ ≤ i ≤ ≤ vertices w (1 i n) is adjacent to the vertices u (1 i n) with edges s (1 i n). i ≤ ≤ i ≤ ≤ i ≤ ≤ By the definition of line graph on K the edges e ,a ,s (1 i n) of K are the 1,n,n,n i i i ≤ ≤ 1,n,n,n vertices of L(K1,n,n,n). Clearly

V [L(K )] = e :1 i n a :1 i n s :1 i n 1,n,n,n { i ≤ ≤ } { i ≤ ≤ } { i ≤ ≤ } [ [ The vertices e (1 i n) induces a clique of order n (say K ) in L(K ). Therefore i ≤ ≤ n 1,n,n,n χ [L(K )] n. = 1,n,n,n ≥ Now consider the vertex set V [L(K )] and the color class C = c ,c , c . 1,n,n,n { 1 2 · · · n}

Assign an equitable coloring to L(K1,n,n,n) by Algorithm 2. Therefore

χ [L(K )] n. = 1,n,n,n ≤ An easy check shows that v v 1. Hence || i| − | j || ≤

χ=[L(K1,n,n,n)] = n. ¾

§5. Equitable Coloring on Middle and Total Graphs of Triple Star Graph

Algorithm 3.

′ Input: The number ‘n of K1,n,n,n;

Output: Assigning equitable coloring for the vertices in M(K1,n,n,n) and T (K1,n,n,n). Equitable Coloring on Triple Star Graph Families 29

begin for i =1 to n

{ V = e ; 1 { i} C(ei)= i; V = s ; 2 { i} C(si)= i;

} V = v ; 3 { } C(v)= n + 1; for i =2 to n

{ V = v ; 4 { i} C(v )= i 1; i − } C(v1)= n; for i =3 to n

{ V = a ; 5 { i} C(a )= i 2; i − } C(a1)= n + 1;

C(a2)= n + 1; for i =4 to n

{ V = w 6 { i} C(w )= i 3; i − } C(w )= n 1; 1 − C(w2)= n; 30 K.Praveena and M.Venkatachalam

C(w3)= n + 1; for i =1 to n

{ V = u ; 7 { i} C(ui)= i + 1;

} V = V V V V V V V 1 ∪ 2 ∪ 3 ∪ 4 ∪ 5 ∪ 6 ∪ 7 end

Theorem 5.1 For any triple star graph K1,n,n,n the equitable chromatic number,

χ [M(K )] = n +1, n 4. = 1,n,n,n ≥

Proof Let V (K )= v v :1 i n w :1 i n u :1 i n . 1,n,n,n { } { i ≤ ≤ }∪ { i ≤ ≤ } { i ≤ ≤ } By the definition of middle graph on K each edge vv , v w and w u (1 i n) in S 1,n,n,n i i iS i i ≤ ≤ K are subdivided by the vertices e , w , s (1 i n) respectively. Clearly 1,n,n,n i i i ≤ ≤ V [M(K )] = v v :1 i n w :1 i n 1,n,n,n { } { i ≤ ≤ } { i ≤ ≤ } u[:1 i n e[:1 i n { i ≤ ≤ } { i ≤ ≤ } [ a :1 i n [ s :1 i n { i ≤ ≤ } { i ≤ ≤ } [ [ The vertices v and e (1 i n) induces a clique of order n+1 (say k ) in [M(K )]. i ≤ ≤ n+1 1,n,n,n Therefore χ [M(K )] n +1. = 1,n,n,n ≥ Now consider the vertex set V [M(K )] and the color class C = c ,c , c . 1,n,n,n { 1 2 · · · n+1} Assign an equitable coloring to M(K1,n,n,n) by Algorithm 3. Therefore

χ M[(K )] n +1, v v 1. = 1,n,n,n ≤ || i| − | j || ≤ Hence

χ [M(K )] = n +1 n 4. ¾ = 1,n,n,n ∀ ≥

Theorem 5.2 For any triple star graph K1,n,n,n the equitable chromatic number,

χ [T (K )] = n +1, n 4. = 1,n,n,n ≥

Proof Let V (K ) = v v : 1 i n w : 1 i n u : 1 i n and 1,n,n,n { } { i ≤ ≤ } { i ≤ ≤ } { i ≤ ≤ } E(K )= e :1 i n a :1 i n s :1 i n . 1,n,n,n { i ≤ ≤ } { Si ≤ ≤ }∪{ iS ≤ ≤ } S S Equitable Coloring on Triple Star Graph Families 31

By the definition of Total graph, the edge vv , v w and w u (1 i n) of K be i i i i i ≤ ≤ 1,n,n,n subdivided by the vertices e , a and s (1 i n) respectively. Clearly i i i ≤ ≤ V [T (K )] = v v :1 i n w :1 i n 1,n,n,n { } { i ≤ ≤ } { i ≤ ≤ } u[:1 i n [e :1 i n { i ≤ ≤ } { i ≤ ≤ } [ a :1 i n [ s :1 i n . { i ≤ ≤ } { i ≤ ≤ } [ [ The vertices v and e (1 i n) induces a clique of order n + 1 (say k ) in T (K ). i ≤ ≤ n+1 1,n,n,n Therefore χ [T (K )] n +1, n 4. = 1,n,n,n ≥ ≥ Now consider the vertex set V (T (K )) and the color class C = c ,c , ,c . 1,n,n,n { 1 2 · · · n+1} Assign an equitable coloring to T (K1,n,n,n) by Algorithm 3. Therefore

χ [T (K )] n +1, n 4, v v 1. = 1,n,n,n ≤ ≥ || i| − | j || ≤ Hence

χ [T (K )] = n +1, n 4. ¾ = 1,n,n,n ∀ ≥

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